Fig. 2.22.3. Variation of the cross-field diffusion coefficient kl versus the perturbation amplitude 5B and the wave propagation anisotropy (angle a) for the flat spectrum and the Kolmogorov spectrum. Results for the Alfven turbulence (thin lines) and the fast-mode turbulence (thick lines with indicated simulation points) are superimposed on the same panels. From Michalek and Ostrowski (1999).
The characteristic features seen in Fig. 2.22.3 can be qualitatively explained with the use of simple physical arguments discussed by Michalek and Ostrowski (1998). Results presented there show much larger increases of respective Dm than kl . It proves that in the range of SB considered here kl is in a substantial degree controlled by the cross-field drifts and the resonance cyclotron scattering, and not by the field line diffusion. Michalek and Ostrowski (1999) stressed that the substantial cross-field shifts accompany wave particle interaction involving the so called 'transit time damping resonance', where for the effective cross-field drift, the particle velocity v// and the wave phase velocity V// along the mean field are approximately equal (V// = va for the Alfven waves and V// = va (k/k//) for the magneto-sonic fast-mode waves). For va = 10-3 c and v = 0.99c considered in the described simulations a noted difference between kl for the Alfven and the fastmode waves occurs as a result of satisfying the resonance condition in a wider range of v// by oblique fast-mode waves. Another difference arises from the linear compressive terms occur only in fast-mode waves. It enables for gradient drifts at small perturbation amplitudes and enables particle cross-field transport when interacting with long waves.
2.23. Dispersion relations for CR particle diffusive propagation
Kota (1999) introduced and evaluated the dispersion relations for CR particles diffusive propagation. He presented illustrative examples for cases including dominant helicity, focusing, and hemispherical scattering. It was shown that the dispersion relations can be quickly computed and can be a useful diagnostic tool for exploring the validity range of various approximations. The matter of the problem is as following.
The evolution of the distribution function f (z,u, t) for CR particle diffusive propagation in time t, space z, and cosine of pitch-angle p is governed in the simplest rectilinear geometry by the equation df df d „ df — + v/u— =— Dm — . dt dz dju dju
where Dju is the pitch-angle scattering coefficient. Eq. 2.23.1 is often approximated by the diffusion equation which operates with the omni-directional density fo(z,t) = (f (z,u,t)) only (< > indicates average over p). The diffusion model is inaccurate for short times and fails to describe the early phase of SEP events. Several efforts have been made to improve the diffusion model. Fisk and Axford (1969) introduced the telegrapher's equation. According to Kota (1999), the modified equations can be written in the general form of adf + Adf + 1 dfo t
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